A101263 -id:A101263 - cp's OEIS frontend

A003401 Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285

Offset: 1

N. J. A. Sloane, R. K. Guy

The terms 1 and 2 correspond to degenerate polygons.
These are also the numbers for which phi(n) is a power of 2: A209229(A000010(a(n))) = 1. - Olivier Gérard Feb 15 1999
From Stanislav Sykora, May 02 2016: (Start)
The sequence can be also defined as follows: (i) 1 is a member. (ii) Double of any member is also a member. (iii) If a member is not divisible by a Fermat prime F_k then its product with F_k is also a member. In particular, the powers of 2 (A000079) are a subset and so are the Fermat primes (A019434), which are the only odd prime members.
The definition is too restrictive (though correct): The Georg Mohr - Lorenzo Mascheroni theorem shows that constructibility using a straightedge and a compass is equivalent to using compass only. Moreover, Jean Victor Poncelet has shown that it is also equivalent to using straightedge and a fixed ('rusty') compass. With the work of Jakob Steiner, this became part of the Poncelet-Steiner theorem establishing the equivalence to using straightedge and a fixed circle (with a known center). A further extension by Francesco Severi replaced the availability of a circle with that of a fixed arc, no matter how small (but still with a known center).
Constructibility implies that when m is a member of this sequence, the edge length 2*sin(Pi/m) of an m-gon with circumradius 1 can be written as a finite expression involving only integer numbers, the four basic arithmetic operations, and the square root. (End)
If x,y are terms, and gcd(x,y) is a power of 2 then x*y is also a term. - David James Sycamore, Aug 24 2024

			34 is a term of this sequence because a circle can be divided into exactly 34 parts. 7 is not.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 183.
Allan Clark, Elements of Abstract Algebra, Chapter 4, Galois Theory, Dover Publications, NY 1984, page 124.
Duane W. DeTemple, "Carlyle circles and the Lemoine simplicity of polygon constructions." The American Mathematical Monthly 98.2 (1991): 97-108. - N. J. A. Sloane, Aug 05 2021
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. L. van der Waerden, Modern Algebra. Unger, NY, 2nd ed., Vols. 1-2, 1953, Vol. 1, p. 187.

Amiram Eldar, Table of n, a(n) for n = 1..10136 (terms below 10^100; terms 1..2000 from T. D. Noe)
Laura Anderson, Jasbir S. Chahal and Jaap Top, The last chapter of the Disquisitiones of Gauss, arXiv:2110.01355 [math.HO], 2021.
Wayne Bishop, How to construct a regular polygon, Amer. Math. Monthly 85(3) (1978), 186-188.
Alessandro Chiodo, A note on the construction of the Śrī Yantra, Sorbonne Université (Paris, France, 2020).
T. Chomette, Construction des polygones réguliers (in French).
Duane W. DeTemple, Carlyle circles and the Lemoine simplicity of polygon constructions, Amer. Math. Monthly 98(2) (1991), 97-108.
Bruce Director, Measurement and Divisibility.
David Eisenbud and Brady Haran, Heptadecagon and Fermat Primes (the math bit), Numberphile video (2015).
Mauro Fiorentini, Construibili (numeri).
C. F. Gauss, Disquisitiones Arithmeticae, 1801. English translation: Yale University Press, New Haven, CT, 1966, p. 463. Original (in Latin).
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag. 63(1) (1990), 3-20. [Annotated scanned copy] [DOI]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Johann Gustav Hermes, Über die Teilung des Kreises in 65537 gleiche Teile (About the division of the circle into 65537 equal pieces), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Vol. 3 (1894), 170-186.
Friedrich Julius Richelot, De resolutione algebraica aequationis X^257 = 1, sive de divisione circuli per bisectionem anguli septies repetitam in partes 257 inter se aequales commentatio coronata (On the resolution of the algebraic equation X^257 = 1, or ...), Journal für die reine und angewandte Mathematik 9 (1832), 1-26.
Pierre Wantzel, Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas (Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass), Journal de Mathématiques Pures et Appliquées 2 (1837), 366-372.
Eric Weisstein's World of Mathematics, Constructible Number.
Eric Weisstein's World of Mathematics, Constructible Polygon.
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Trigonometry.
Eric Weisstein's World of Mathematics, Trigonometry Angles.
Wikipedia, Constructible polygon.
Wikipedia, Johann Gustav Hermes.
Wikipedia, Friedrich Julius Richelot.
Wikipedia, Mohr-Mascheroni theorem.
Wikipedia, Pierre Wantzel.
Wikipedia, Poncelet-Steiner theorem.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.

Subsequence of A295298. - Antti Karttunen, Nov 27 2017
A004729 and A051916 are subsequences. - Reinhard Zumkeller, Mar 20 2010
Cf. A000079, A004169, A000215, A099884, A019434 (Fermat primes).
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17), A272536 (20).
Positions of zeros in A293516 (apart from two initial -1's), and in A336469, positions of ones in A295660 and in A336477 (characteristic function).
Cf. also A046528.
Cf. A000079, A092506.

a003401 n = a003401_list !! (n-1)
a003401_list = map (+ 1) $ elemIndices 1 $ map a209229 a000010_list
-- Reinhard Zumkeller, Jul 31 2012

Select[ Range[ 1300 ], IntegerQ[ Log[ 2, EulerPhi[ # ] ] ]& ] (* Olivier Gérard Feb 15 1999 *)
(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Take[ Union[ Flatten[ NestList[2# &, Times @@@ Table[ UnrankSubset[n, Join[{1}, Table[2^2^i + 1, {i, 0, 4}]]], {n, 63}], 11]]], 60] (* Robert G. Wilson v, Jun 11 2005 *)
nn=10; logs=Log[2,{2,3,5,17,257,65537}]; lim2=Floor[nn/logs[[1]]]; Sort[Reap[Do[z={i,j,k,l,m,n}.logs; If[z<=nn, Sow[2^z]], {i,0,lim2}, {j,0,1}, {k,0,1}, {l,0,1}, {m,0,1}, {n,0,1}]][[2,1]]]
A092506 = {2, 3, 5, 17, 257, 65537}; s = Sort[Times @@@ Subsets@ A092506]; mx = 1300; Union@ Flatten@ Table[(2^n)*s[[i]], {i, 64}, {n, 0, Log2[mx/s[[i]]]}] (* Robert G. Wilson v, Jul 28 2014 *)

for(n=1,10^4,my(t=eulerphi(n));if(t/2^valuation(t,2)==1,print1(n,", "))); \\ Joerg Arndt, Jul 29 2014

is(n)=n>>=valuation(n,2); if(n<7, return(n>0)); my(k=logint(logint(n,2),2)); if(k>32, my(p=2^2^k+1); if(n%p, return(0)); n/=p; unknown=1; if(n%p==0, return(0)); p=0; if(is(n)==0, 0, "unknown [has large Fermat number in factorization]"), 4294967295%n==0) \\ Charles R Greathouse IV, Jan 09 2022

is(n)=n>>=valuation(n,2); 4294967295%n==0 \\ valid for n <= 2^2^33, conjecturally valid for all n; Charles R Greathouse IV, Jan 09 2022

from sympy import totient
A003401_list = [n for n in range(1,10**4) if format(totient(n),'b').count('1') == 1]
# Chai Wah Wu, Jan 12 2015

Terms from 3 onward are computable as numbers such that cototient-of-totient equals the totient-of-totient: Flatten[Position[Table[co[eu[n]]-eu[eu[n]], {n, 1, 10000}], 0]] eu[m]=EulerPhi[m], co[m]=m-eu[m]. - Labos Elemer, Oct 19 2001, clarified by Antti Karttunen, Nov 27 2017
Any product of 2^k and distinct Fermat primes (primes of the form 2^(2^m)+1). - Sergio Pimentel, Apr 30 2004, edited by Franklin T. Adams-Watters, Jun 16 2006
If the well-known conjecture that there are only five prime Fermat numbers F_k=2^{2^k}+1, k=0,1,2,3,4 is true, then we have exactly: Sum_{n>=1} 1/a(n)= 2*Product_{k=0..4} (1+1/F_k) = 4869735552/1431655765 = 3.40147098978.... - Vladimir Shevelev and T. D. Noe, Dec 01 2010
log a(n) >> sqrt(n); if there are finitely many Fermat primes, then log a(n) ~ k log n for some k. - Charles R Greathouse IV, Oct 23 2015

Definition clarified by Bill Gosper. - N. J. A. Sloane, Jun 14 2020

A091999 Numbers that are congruent to {2, 10} mod 12.

Original entry on oeis.org

2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298, 302, 310, 314, 322, 326, 334

Offset: 1

Ray Chandler, Feb 21 2004

Numbers divisible by 2 but not by 3 or 4. - Robert Israel, Apr 24 2015
For n > 1, a(n) is representable as a sum of four but no fewer consecutive nonnegative integers, i.e., 10 = 1 + 2 + 3 + 4, 14 = 2 + 3 + 4 + 5, 22 = 4 + 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016
Essentially the same as A063221. - Omar E. Pol, Aug 16 2023

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Second row of A092260.
Cf. A109761 (subsequence).
Cf. A002193, A007310, A101263, A138591, A186424.
Cf. A017545, A017641, A063221.

a091999 n = a091999_list !! (n-1)
a091999_list = 2 : 10 : map (+ 12) a091999_list
-- Reinhard Zumkeller, Jan 21 2013

[6*n-3+(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Apr 23 2015

A091999:=n->6*n-3+(-1)^n: seq(A091999(n), n=1..100); # Wesley Ivan Hurt, Apr 23 2015

Flatten[#+{2,10}&/@(12*Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{2,10,14},60] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 6*n - 3 + (-1)^n \\ David Lovler, Jul 16 2022

a(n) = 2*A007310(n).
a(n) = A186424(n) - A186424(n-2), for n > 1.
a(n) = 12*(n-1) - a(n-1), with a(1)=2. - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = a(n-1) + a(n-2) - a(n-3); a(1)=2, a(2)=10, a(3)=14. - Harvey P. Dale, Jun 24 2013
a(n) = 6*n - 3 + (-1)^n. - Wesley Ivan Hurt, Apr 23 2015
E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - Stefano Spezia, May 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)). - Amiram Eldar, Dec 13 2021
E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - David Lovler, Aug 08 2022
a(n) = A063221(n), n > 1. - Omar E. Pol, Aug 15 2023
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2) (A002193).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/12) (A101263). (End)

A019824 Decimal expansion of sine of 15 degrees.

Original entry on oeis.org

2, 5, 8, 8, 1, 9, 0, 4, 5, 1, 0, 2, 5, 2, 0, 7, 6, 2, 3, 4, 8, 8, 9, 8, 8, 3, 7, 6, 2, 4, 0, 4, 8, 3, 2, 8, 3, 4, 9, 0, 6, 8, 9, 0, 1, 3, 1, 9, 9, 3, 0, 5, 1, 3, 8, 1, 4, 0, 0, 3, 2, 0, 7, 3, 1, 5, 0, 5, 6, 9, 7, 4, 7, 4, 8, 8, 0, 1, 9, 9, 6, 9, 2, 2, 3, 6, 7, 9, 7, 4, 6, 9, 4, 2, 4, 9, 6, 6, 5

Offset: 0

N. J. A. Sloane

Also the imaginary part of i^(1/6). - Stanislav Sykora, Apr 25 2012

			0.258819045102520762348898837624048328349068901319930513814003207315...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mark B. Villarino, Legendre's Singular Modulus, arXiv:2002.07250 [math.HO], 2020.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 4

Cf. A002194, A020765, A101263.

RealDigits[ Sin[ Pi/12], 10, 111][[1]] (* Or *) RealDigits[(Sqrt[3] - 1)/(2 Sqrt[2]), 10, 111][[1]] (* Robert G. Wilson v *)
RealDigits[Sin[15 Degree],10,120][[1]] (* Harvey P. Dale, Jul 16 2016 *)

sin(Pi/12) \\ Charles R Greathouse IV, Apr 25 2012

Equals (sqrt(3)-1)/(2*sqrt(2)) = (A002194 -1) * A020765 = sin(Pi/12). - R. J. Mathar, Jun 18 2006
Equals 2F1(9/8,-1/8;1/2;3/4) / 2 = - 2F1(11/8,-3/8;1/2;3/4) / 2 = cos(5*Pi/12). - R. J. Mathar, Oct 27 2008
Equals sqrt(2 - sqrt(3))/2 = (1/2) * A101263. - Amiram Eldar, Aug 05 2020
Equals A019819 * A019894 + A019814 * A019889 = A019821 * A019896 + A019812 * A019887. - R. J. Mathar, Jan 27 2021
This^2 + A019884^2=1. - R. J. Mathar, Aug 31 2025
Smallest positive of the 4 real-valued roots of 16*x^4-16*x^2+1=0. - R. J. Mathar, Aug 31 2025

A188887 Decimal expansion of sqrt(2 + sqrt(3)).

Original entry on oeis.org

1, 9, 3, 1, 8, 5, 1, 6, 5, 2, 5, 7, 8, 1, 3, 6, 5, 7, 3, 4, 9, 9, 4, 8, 6, 3, 9, 9, 4, 5, 7, 7, 9, 4, 7, 3, 5, 2, 6, 7, 8, 0, 9, 6, 7, 8, 0, 1, 6, 8, 0, 9, 1, 0, 0, 8, 0, 4, 6, 8, 6, 1, 5, 2, 6, 2, 0, 8, 4, 6, 4, 2, 7, 9, 5, 9, 7, 1, 1, 0, 3, 2, 6, 9, 5, 1, 2, 3, 4, 8, 3, 7, 1, 6, 1, 4, 0, 9, 0, 3, 7, 7, 6, 8, 0, 4, 2, 2, 3, 7, 2, 8, 7, 6, 3, 2, 4, 3, 0, 7, 4, 8, 9, 1, 8, 5, 0, 7, 5, 7

Offset: 1

Clark Kimberling, Apr 12 2011

Decimal expansion of the length/width ratio of a sqrt(2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A sqrt(2)-extension rectangle matches the continued fraction [1,1,13,1,2,15,10,1,18,1,1,21,,...] (A188888) for the shape L/W = sqrt(2 + sqrt(3)). This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...].  Specifically, for the sqrt(2)-extension rectangle, 1 square is removed first, then 1 square, then 13 squares, then 1 square, ..., so that the original rectangle of shape sqrt(2 + sqrt(3)) is partitioned into an infinite collection of squares.
sqrt(2 + sqrt(3)) is also the shape of the greater sqrt(6)-contraction rectangle; see A188738.
This constant is also the length of the Steiner span of three vertices of a unit square. - Jean-François Alcover, May 22 2014
It is also the larger positive coordinate of (symmetrical) intersection points created by x^2 + y^2 = 4 circle and y = 1/x hyperbola. The smaller coordinate is A101263. - Leszek Lezniak, Sep 18 2018
Length of the shortest diagonal in a regular 12-gon with unit side. - Mohammed Yaseen, Nov 12 2020

			1.931851652578136573499486399457794735267809678016809...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Burkard Polster, Irrational roots, Mathologer video (2018).
Index entries for algebraic numbers, degree 4.

Cf. A019884, A019973, A091998, A101263, A188640, A188888, A214726, A217870, A329247.

Sqrt(2 + Sqrt(3)); // G. C. Greubel, Apr 10 2018

r = 2^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
ContinuedFraction[t, 120]
RealDigits[Sqrt[2 + Sqrt[3]], 10, 100][[1]] (* G. C. Greubel, Apr 10 2018 *)

sqrt(2 + sqrt(3)) \\ G. C. Greubel, Apr 10 2018

Equals (sqrt(6) + sqrt(2))/2.
Equals exp(asinh(cos(Pi/4))). - Geoffrey Caveney, Apr 23 2014
Equals cos(Pi/4) + sqrt(1 + cos(Pi/4)^2). - Geoffrey Caveney, Apr 23 2014
Equals i^(1/6) + i^(-1/6). - Gary W. Adamson, Jul 07 2022
Equals the largest root of x - 1/x = sqrt(2) and of x^2 + 1/x^2 = 4. - Gary W. Adamson, Jun 12 2023
Equals Product_{k>=0} ((12*k + 2)*(12*k + 10))/((12*k + 1)*(12*k + 11)). - Antonio Graciá Llorente, Feb 24 2024
From Amiram Eldar, Nov 23 2024: (Start)
Equals A214726 / 2 = 2 * A019884 = 1 / A101263 = exp(A329247) = A217870^2 = sqrt(A019973).
Equals Product_{k>=1} (1 - (-1)^k/A091998(k)). (End)

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

Original entry on oeis.org

1, 0, 3, 5, 2, 7, 6, 1, 8, 0, 4, 1, 0, 0, 8, 3, 0, 4, 9, 3, 9, 5, 5, 9, 5, 3, 5, 0, 4, 9, 6, 1, 9, 3, 3, 1, 3, 3, 9, 6, 2, 7, 5, 6, 0, 5, 2, 7, 9, 7, 2, 2, 0, 5, 5, 2, 5, 6, 0, 1, 2, 8, 2, 9, 2, 6, 0, 2, 2, 7, 8, 9, 8, 9, 9, 5, 2, 0, 7, 9, 8, 7, 6, 8, 9, 4, 7, 1, 8, 9, 8, 7, 7, 6, 9, 9, 8, 6, 6, 2, 0, 8, 3, 5, 8

Offset: 1

Rick L. Shepherd, Jun 24 2006

Side length of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link).

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			1.03527618041008304939559535049619331339627560527972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Eric Weisstein's World of Mathematics, Equilateral Triangle.
Index entries for algebraic numbers, degree 4.

Cf. A002193, A019679, A019691, A019884, A010464, A092242, A101263.

RealDigits[Sec[15 Degree],10,120][[1]] (* Harvey P. Dale, Jun 03 2015 *)

sqrt(6) - sqrt(2) \\ Charles R Greathouse IV, Aug 27 2017

Equals sec(Pi/12) = sec(A019679) = sqrt(6) - sqrt(2) = A010464 - A002193 = csc(5*Pi/12) = 1/sin(5*Pi/12) = 1/sin(10*A019691) = 1/A019884.
Equals Product_{k >= 1} 1/(1 - 1/(36*(2*k - 1)^2)). - Antonio Graciá Llorente, Mar 20 2024
From Amiram Eldar, Nov 24 2024: (Start)
Equals 2*A101263.
Equals Product_{k>=1} (1 - (-1)^k/A092242(k)). (End)
Smallest positive of the 4 real-valued roots of x^4-16*x^2+16=0. - R. J. Mathar, Aug 31 2025

A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

Original entry on oeis.org

4, 1, 5, 8, 2, 3, 3, 8, 1, 6, 3, 5, 5, 1, 8, 6, 7, 4, 2, 0, 3, 4, 8, 4, 5, 6, 8, 8, 1, 0, 2, 5, 0, 3, 3, 2, 4, 3, 3, 1, 6, 9, 5, 2, 1, 2, 5, 5, 4, 4, 7, 6, 7, 2, 8, 1, 4, 3, 6, 3, 9, 4, 7, 7, 6, 4, 7, 6, 5, 6, 5, 1, 3, 2, 8, 1, 4, 8, 7, 5, 2, 6, 0, 9, 2, 5, 7, 5, 1, 3, 4, 4, 5, 4, 5, 5, 1, 4, 6, 1, 1, 5, 7, 3, 0

Offset: 0

Stanislav Sykora, May 02 2016

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.
This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

			0.415823381635518674203484568810250332433169521255447672814363947...

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for sequences related to Chebyshev polynomials.
Index entries for algebraic numbers, degree 8.

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/15)
```

Equals 2*sin(Pi/m) for m=15, 2*A019821.
Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.
Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018

A108412 Expansion of (1 + x + x^2)/(1 - 4x^2 + x^4).

Original entry on oeis.org

1, 1, 5, 4, 19, 15, 71, 56, 265, 209, 989, 780, 3691, 2911, 13775, 10864, 51409, 40545, 191861, 151316, 716035, 564719, 2672279, 2107560, 9973081, 7865521, 37220045, 29354524, 138907099, 109552575, 518408351, 408855776, 1934726305

Offset: 0

Ralf Stephan, Jun 05 2005

This is the sequence of Lehmer numbers u_n(sqrt(R),Q) with the parameters R = 6 and Q = 1. It is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence satisfies a linear recurrence of order four. - Peter Bala, Apr 18 2014
The sequence of convergents of the 2-periodic continued fraction [0; 1, -6, 1, -6, ...] = 1/(1 - 1/(6 - 1/(1 - 1/(6 - ...)))) = 3 - sqrt(3) begins [0/1, 1/1, 6/5, 5/4, 24/19, 19/15, 90/71,...]. The present sequence is the sequence of denominators; the sequence of numerators of the continued fraction convergents [1, 6, 5, 24, 19, 90,...] is also a strong divisibility sequence. Cf. A005013 and A203976. - Peter Bala, May 19 2014
From Peter Bala, Mar 25 2018: (Start)
The following remarks assume an offset of 1.
Define a binary operation o on the real numbers by x o y = x*sqrt(1 + (1/2)*y^2) + y*sqrt(1 + (1/2)*x^2). The operation o is commutative and associative with identity 0. We have a(2*n + 1) = 1 o 1 o ... o 1 (2*n + 1 terms) and sqrt(6)*a(2*n) = (1 o 1 o ... o 1) (2*n terms). Cf. A005013 and A084068. For example, 1 o 1 = sqrt(6) and 1 o 1 o 1 = sqrt(6) o 1 = 5 = a(3).
From the obvious identity ( 1 o 1 o ... o 1 (2*n terms) ) o ( 1 o 1 o ... o 1 (2*m terms) ) = 1 o 1 o ... o 1 (2*n + 2*m terms) we find the relation a(2*n+2*m) = a(2*n)*sqrt(1 + 3*a(2*m)^2) + a(2*m)*sqrt(1 + 3*a(2*n)^2).
Similarly, from a(2*n+1) o a(2*m+1) = sqrt(6)*a(2*n+2*m+2) we find sqrt(6)*a(2*n+2*m+2) = a(2*n+1)*sqrt(1 + (1/2)*a(2*m+1)^2) + a(2*m+1)*sqrt(1 + (1/2)*a(2*n+1)^2). (End)

			G.f. = 1 + x + 5*x^2 + 4*x^3 + 19*x^4 + 15*x^5 + 71*x^6 + 56*x^7 + ...

Harvey P. Dale, Table of n, a(n) for n = 0..1000
P. Bala, Notes on 2-periodic continued fractions and Lehmer sequences
Seong Ju Kim, R. Stees, L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
E. W. Weisstein, MathWorld: Lehmer Number
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A001353, A001834.

Cf. A026741, A005013, A084068.

a := proc (n) if `mod`(n, 2) = 1 then 1/sqrt(2)*( ((sqrt(6) + sqrt(2))/2 )^n - ( (sqrt(6) - sqrt(2))/2 )^n) else 1/sqrt(12)*( ((sqrt(6) + sqrt(2))/2 )^n - ( (sqrt(6) - sqrt(2))/2 )^n) end if;
end proc:
seq(simplify(a(n)), n = 1..30); # Peter Bala, Mar 25 2018

CoefficientList[Series[(1+x+x^2)/(1-4x^2+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,4,0,-1},{1,1,5,4},40] (* Harvey P. Dale, Nov 15 2012 *)

{a(n) = my( w = quadgen(24)); simplify( polchebyshev( n, 2, w/2) / if( n%2, w, 1))}; /* Michael Somos, Feb 10 2015 */

a(0)=a(1)=1, a(2)=5, a(n)a(n+3) - a(n+1)a(n+2) = -1.
a(0)=1, a(1)=1, a(2)=5, a(3)=4, a(n) = 4*a(n-2)-a(n-4). - Harvey P. Dale, Nov 15 2012
a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, and a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even, where alpha = (1/2)*(sqrt(6) + sqrt(2)) (A188887) and beta = (1/2)*(sqrt(6) - sqrt(2)) (A101263). Equivalently, a(n) = U(n-1,sqrt(6)/2) for n odd and a(n) = (1/sqrt(6))*U(n-1,sqrt(6)/2) for n even, where U(n,x) is the Chebyshev polynomial of the second kind. - Peter Bala, Apr 18 2014
a(2*n) = A001834(n). a(2*n + 1) = A001353(n+1). - Michael Somos, Feb 10 2015
a(n) = -a(-2-n) for all n in Z. - Michael Somos, Feb 10 2015

A272535 Decimal expansion of the edge length of a regular 16-gon with unit circumradius.

Original entry on oeis.org

3, 9, 0, 1, 8, 0, 6, 4, 4, 0, 3, 2, 2, 5, 6, 5, 3, 5, 6, 9, 6, 5, 6, 9, 7, 3, 6, 9, 5, 4, 0, 4, 4, 4, 8, 1, 8, 5, 5, 3, 8, 3, 2, 3, 5, 5, 0, 3, 9, 0, 9, 6, 1, 5, 5, 0, 9, 0, 0, 4, 1, 7, 8, 9, 8, 9, 5, 2, 6, 6, 3, 7, 5, 7, 1, 8, 4, 9, 1, 6, 0, 4, 5, 0, 6, 5, 0, 6, 1, 8, 4, 6, 8, 1, 8, 0, 7, 6, 3, 4, 6, 1, 9, 8, 4

Offset: 0

Stanislav Sykora, May 02 2016

Like all m-gons with m equal to a power of 2 (see A003401 and A000079), this is a constructible number.

			0.390180644032256535696569736954044481855383235503909615509004...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A000079, A003401.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/16], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/16)
```

Equals 2*sin(Pi/m) for m=16, 2*A232738. Equals also sqrt(2-sqrt(2+sqrt(2))).

A272536 Decimal expansion of the edge length of a regular 20-gon with unit circumradius.

Original entry on oeis.org

3, 1, 2, 8, 6, 8, 9, 3, 0, 0, 8, 0, 4, 6, 1, 7, 3, 8, 0, 2, 0, 2, 1, 0, 6, 3, 8, 9, 3, 4, 3, 3, 3, 7, 8, 4, 6, 2, 7, 7, 9, 9, 7, 8, 4, 1, 7, 1, 3, 2, 1, 5, 8, 0, 1, 6, 9, 2, 8, 2, 6, 9, 2, 1, 1, 5, 5, 1, 7, 5, 8, 6, 6, 1, 1, 2, 4, 7, 1, 5, 8, 6, 7, 3, 3, 9, 1, 7, 4, 5, 3, 5, 3, 6, 9, 7, 3, 7, 6, 7, 5, 0, 2, 8, 0

Offset: 0

Stanislav Sykora, May 02 2016

Since 20-gon is constructible (see A003401), this is a constructible number.

			0.3128689300804617380202106389343337846277997841713215801692826921...

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for algebraic numbers, degree 8.

Cf. A003401.
Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272534 (15), A272535 (16), A228787 (17).
Cf. A019818.

RealDigits[N[2Sin[Pi/20], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/20)
```

Equals 2*sin(Pi/20) = 2*A019818.
Equals also (sqrt(2)+sqrt(10)-2*sqrt(5-sqrt(5)))/4.
Equals i^(9/10) + i^(-9/10). - Gary W. Adamson, Jul 08 2022

A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

Original entry on oeis.org

4, 2, 1, 2, 7, 9, 5, 4, 3, 9, 8, 3, 9, 0, 3, 4, 3, 2, 7, 6, 8, 8, 2, 1, 7, 6, 0, 6, 5, 0, 2, 9, 8, 0, 9, 1, 6, 1, 0, 3, 6, 7, 2, 1, 4, 0, 7, 2, 6, 1, 2, 2, 3, 2, 1, 6, 5, 4, 3, 7, 5, 4, 5, 4, 0, 6, 5, 1, 7, 2, 9, 3, 9, 2, 2, 4, 3, 7, 7, 9, 1, 5, 3, 6, 3, 2, 9, 0, 6, 8, 8, 4, 7, 1, 9, 2, 4, 6, 2, 4, 3, 9

Offset: 0

Jeremy Tan, Jan 14 2017

The corresponding values for two to nine points have simple expressions:
  N ... d_min
  2 ... sqrt(2)            (A002193)
  3 ... sqrt(6) - sqrt(2)  (A120683)
  4 ... 1                  (A000007)
  5 ... sqrt(2) / 2        (A010503)
  6 ... sqrt(13) / 6       (A381485)
  7 ... 4 - 2*sqrt(3)      (A379338)
  8 ... sqrt(2 - sqrt(3))  (A101263)
  9 ... 1 / 2              (A020761)
In contrast, the value for ten points has a minimal polynomial of degree 18.
The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...

			0.421279543983903432768821760650298...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

C. de Groot, R. Peikert, and D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, ETH Zürich, No. 90-12, August 1990.
Eckard Specht, The best known packings of equal circles in a square.
Jeremy Tan, Sympy (Python) program.
Index entries for algebraic numbers, degree 18.

Cf. A281115 (10 points in unit circle), A000007, A002193, A010503, A020761, A101263, A120683, A379338, A381485.

RealDigits[x /. FindRoot[x^Range[18, 0, -1].{1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200}, {x, 2/5}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 24 2025 *)

my(p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200])); polrootsreal(p)[1]

d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.

cp's OEIS Frontend

A003401 Numbers of edges of regular polygons constructible with ruler (or, more precisely, an unmarked straightedge) and compass.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

Python

Formula

Extensions

A091999 Numbers that are congruent to {2, 10} mod 12.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

A019824 Decimal expansion of sine of 15 degrees.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A188887 Decimal expansion of sqrt(2 + sqrt(3)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs